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Nonlinear eigenvalue problems arising from nonlinear fracture mechanics analysis E.M. Adulina, L.V. Stepanova, Samara State University

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J. W. Hutchinson, Singular behavior at the end of tensile crack in a hardening material, J. Mech. Phys. Solids 16 (1968) 13-31. J. R. Rice, G. F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material, J. Mech. Phys. Solids 16 (1968) 1-12. F.G. Yuan, S. Yang, Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear, Int. J. of Fracture 69, (1994) 1-26. G.P. Nikishkov, An algorithm and a computer program for the three-term asymptotic expansion of elastic-plastic crack tip stress and displacement fields, Engineering Fracture Mechanics 50 (1995) 65-83. B.N. Nguyen, P.R. Onck, E. Van Der Giessen, On higher-order crack-tip fields in creeping solids, Transaction of the ASME 67 (2000) 372-382. I. Jeon, S. Im, The role of higher order eigenfields in elastic-plastic cracks, J. Mech. Phys. Solids 49 (2001) 2789-2818. C.Y. Hui, A. Ruina, Why K? High order singularities and small scale yielding, Int. J. of Fracture 72 (1995) 97-120. L. Meng, S.B. Lee, Eigenspectra and orders of singularity at a crack tip for a power-law creeping medium, Int. J. of Fracture 92 (1998) 55-70. M. Anheuser, D. Gross, Higher order fields at crack and notch tips in power-law materials under longitudinal shear, Archive of Applied Mechanics 64 (1994) 509-518.

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Hutchinson-Rice-Rosengren solution

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Crack tip geometry and coordinate systems

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Mode I crack. Basic equations

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The Airy stress potential The asymptotic behavior of the Airy function The asymptotic behavior of the stress field

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Nonlinear eigenvalue problem

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Perturbation theory approach

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Mode III crack. Basic equations

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The stress function The asymptotic behavior of the stress function Nonlinear eigenvalue problem

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Eigenvalues The set of linear differential equations

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The solvability condition The set of boundary value problems The solvability condition

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Closed form analytical solution

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Mode I crack. Nonlinear eigenvalue problem

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The perturbation method The undisturbed linear problem

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The set of boundary value problems

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The solvability condition

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The linear differential equation for

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The three-term asymptotic expansions of the hardening exponent

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The three-term asymptotic expansions for the hardening exponent

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Eigenspectra at a Mode II crack under plane strain conditions The nine-term asymptotic expansion of the hardening exponent

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Eigenspectra at a Mode II crack under plane stress conditions The nine-term asymptotic expansion of the hardening exponent

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The asymptotic study of fatigue crack growth based on continuum damage mechanics Zhao J., Zhang X. The asymptotic study of fatigue crack growth based on damage mechanics// Engn. Fracture Mechanics. 1995. V. 50. 1. P. 131-141. Li J., Recho N. Methodes asymptotiques en mecanique de la rupture. Paris: Hermes Science Publications, 2002. 262 p. Astafiev V.I., Radayev Y.N., Stepanova L.V. Nonlinear fracture mechanics. Samara: Samara State University, 2001. 632 p. Stepanova L.V. Mathematical methods of fracture mechanics. Moscow: Fizmatlit, 2009. 332 p. Astafjev V.I., Grigorova T.V., Pastuchov V.A. Influence of continuum damage on stress distribution near a tip of a growing crack under creep conditions/ procedings of the International Colloquium Mechanics of creep brittle materials 2, University of Leicester, UK, 1991. P. 49-61.

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The asymptotic study of fatigue crack growth based on continuum damage mechanics Consider a fatigue growing crack lying on the x-axis with the coordinate origin located at the moving crack tip

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The asymptotic study of fatigue crack growth based on continuum damage mechanics The essence of continuum damage mechanics is characterized by material deterioration coupled constitutive equations. Under the assumption of linear elasticity a stiffness reduction based stress-strain relationship is applied as The kinetic law of damage evolution The equilibrium equations

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The asymptotic study of fatigue crack growth based on continuum damage mechanics The compatibility equation The constitutive equations The constitutive equations for plane stress conditions The constitutive equations for plane strain conditions

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The asymptotic study of fatigue crack growth based on continuum damage mechanics The traditional traction free conditions on crack surfaces The Airy stress function The Airy stress function can be presented in the form The Mode I stress field components at the crack tip behave as follows

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The asymptotic study of fatigue crack growth based on continuum damage mechanics The damage field around the crack tip can be presented as The strain components are given as From the compatibility equation one can obtain (for plane stress) one can obtain (for plane strain)

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The asymptotic study of fatigue crack growth based on continuum damage mechanics The damage evolution law allows to obtain The symmetry of the stress and damage fields around the crack tip The normalizing condition is chosen as The regularity requirement The traction free conditions

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The asymptotic study of fatigue crack growth based on continuum damage mechanics The totally damaged zone needs to be modeled in the vicinity of the crack tip The analytical result The stress and damage fields around the crack tip

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The asymptotic study of fatigue crack growth based on continuum damage mechanics

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The new analytical presentation of stress, strain and continuity fields both for plane strain and plane stress conditions is given. The results obtained differ from Zhao and Zhang's solution where the original formulation of the problem for plane stress conditions has been proposed. An analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. The perturbation technique is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law.